Input-to-state stabilization of endemic models for uncertainty of transmission, inflows, and immunity waning

Abstract

To mitigate the impact of infectious diseases spread, feedback decision that effectively adjusts the control amount making use of current data has been anticipated. In general, feedback control design is based on models, which are inherently subject to inaccuracy and uncertainty. Control theory seeks robustness guarantees that do not rely on the model perfection which is usually required for prediction purposes. To provide such a key, this paper deals with uncertainty of disease transmission and waning immunity as well as uncertain inflows from neighboring regions. New feedback control laws are proposed to achieve robustness in the framework of input-to-state stabilization (ISS) by governing societal activity levels, vaccination, and isolation. The control addresses endemic situations, which are more practical and mathematically much harder than disease-free situations. To go beyond Jacobian linearization and local analysis, the proposed control covers the entire space of population variables by articulating the achievable globalness mathematically. The preceding ISS-based studies cannot cope with waning immunity no matter how small the waning rate is since it gives rise to supply and dissipation in different growth orders in their formulation. This paper demonstrates how a Lyapunov function and control laws can be constructed to coordinate the orders.